3.51 \(\int \frac{x^4}{\cosh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=73 \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{9 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac{5 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

[Out]

-((x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[a*x]]/(8*a^5) + (9*CoshIntegral[
3*ArcCosh[a*x]])/(16*a^5) + (5*CoshIntegral[5*ArcCosh[a*x]])/(16*a^5)

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Rubi [A]  time = 0.0643322, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5666, 3301} \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{9 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac{5 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCosh[a*x]^2,x]

[Out]

-((x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[a*x]]/(8*a^5) + (9*CoshIntegral[
3*ArcCosh[a*x]])/(16*a^5) + (5*CoshIntegral[5*ArcCosh[a*x]])/(16*a^5)

Rule 5666

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{\cosh ^{-1}(a x)^2} \, dx &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{a \cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{8 x}-\frac{9 \cosh (3 x)}{16 x}-\frac{5 \cosh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{a \cosh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{a \cosh ^{-1}(a x)}+\frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{9 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac{5 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}\\ \end{align*}

Mathematica [A]  time = 0.208964, size = 101, normalized size = 1.38 \[ \frac{-16 a^5 x^5 \sqrt{\frac{a x-1}{a x+1}}-16 a^4 x^4 \sqrt{\frac{a x-1}{a x+1}}+2 \cosh ^{-1}(a x) \text{Chi}\left (\cosh ^{-1}(a x)\right )+9 \cosh ^{-1}(a x) \text{Chi}\left (3 \cosh ^{-1}(a x)\right )+5 \cosh ^{-1}(a x) \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5 \cosh ^{-1}(a x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcCosh[a*x]^2,x]

[Out]

(-16*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] - 16*a^5*x^5*Sqrt[(-1 + a*x)/(1 + a*x)] + 2*ArcCosh[a*x]*CoshIntegral[
ArcCosh[a*x]] + 9*ArcCosh[a*x]*CoshIntegral[3*ArcCosh[a*x]] + 5*ArcCosh[a*x]*CoshIntegral[5*ArcCosh[a*x]])/(16
*a^5*ArcCosh[a*x])

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Maple [A]  time = 0.037, size = 83, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ( -{\frac{1}{8\,{\rm arccosh} \left (ax\right )}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\it Chi} \left ({\rm arccosh} \left (ax\right ) \right ) }{8}}-{\frac{3\,\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{16\,{\rm arccosh} \left (ax\right )}}+{\frac{9\,{\it Chi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{16}}-{\frac{\sinh \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{16\,{\rm arccosh} \left (ax\right )}}+{\frac{5\,{\it Chi} \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{16}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccosh(a*x)^2,x)

[Out]

1/a^5*(-1/8/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/8*Chi(arccosh(a*x))-3/16/arccosh(a*x)*sinh(3*arccosh(a*
x))+9/16*Chi(3*arccosh(a*x))-1/16/arccosh(a*x)*sinh(5*arccosh(a*x))+5/16*Chi(5*arccosh(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} x^{7} - a x^{5} +{\left (a^{2} x^{6} - x^{4}\right )} \sqrt{a x + 1} \sqrt{a x - 1}}{{\left (a^{3} x^{2} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )} + \int \frac{5 \, a^{5} x^{8} - 10 \, a^{3} x^{6} + 5 \, a x^{4} +{\left (5 \, a^{3} x^{6} - 3 \, a x^{4}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} +{\left (10 \, a^{4} x^{7} - 13 \, a^{2} x^{5} + 4 \, x^{3}\right )} \sqrt{a x + 1} \sqrt{a x - 1}}{{\left (a^{5} x^{4} +{\left (a x + 1\right )}{\left (a x - 1\right )} a^{3} x^{2} - 2 \, a^{3} x^{2} + 2 \,{\left (a^{4} x^{3} - a^{2} x\right )} \sqrt{a x + 1} \sqrt{a x - 1} + a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^2,x, algorithm="maxima")

[Out]

-(a^3*x^7 - a*x^5 + (a^2*x^6 - x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x
- a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) + integrate((5*a^5*x^8 - 10*a^3*x^6 + 5*a*x^4 + (5*a^3*x^6 - 3*a
*x^4)*(a*x + 1)*(a*x - 1) + (10*a^4*x^7 - 13*a^2*x^5 + 4*x^3)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^5*x^4 + (a*x +
1)*(a*x - 1)*a^3*x^2 - 2*a^3*x^2 + 2*(a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1
)*sqrt(a*x - 1))), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{\operatorname{arcosh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^4/arccosh(a*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{acosh}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acosh(a*x)**2,x)

[Out]

Integral(x**4/acosh(a*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arcosh}\left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^4/arccosh(a*x)^2, x)